Ind. Eng. Chem. Res. 2006, 45, 6765-6769
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GENERAL RESEARCH Simultaneous Prediction of the Critical and Subcritical Phase Behavior in Mixtures of Perfluoromethane (1)-Alkanes (2) Ilya Polishuk* Department of Chemical Engineering and Biotechnology, the College of Judea and Samaria, 44837 Ariel, Israel
The present study compares the ability of two semi-predictive approaches, namely the Global Phase Diagram approach (GPDA) and predictive Soave-Redlich-Kwong (PSRK), to describe the experimental data in the binary homologous series of perfluoromethane-alkanes. Although this time PSRK is able to describe the patterns of phase behavior in the series under consideration, it fails to predict the data quantitatively. In particular, it underestimates the liquid-liquid equilibria (LLE) range for the light members of the homologous series and overestimates it for the heavy members. In very asymmetric systems, the predictions of PSRK are in total disagreement with the experimental data. In addition, because PSRK does not use the binary parameter l12, it generates an incorrect slope of the LLE critical lines and isopleths in pressure-temperature (P-T) projection. In contrast, GPDA predicts robust results, even for very asymmetric systems, and yields precise description of the more-symmetric ones. These facts confirm a doubtless advantage of the approach that is based on consideration of the complete phase diagram and not only of its separate parts. 1. Introduction Thermodynamic properties of the perfluoromethane (1)alkane (2) systems present significant theoretical interest, because these systems exhibit an unusually large positive deviation from ideality, which results in a strong tendency to exhibit liquid-liquid immiscibility. Thus, although Type IV phase behavior, according to the classification of van Konynenburg and Scott,1 usually occurs in asymmetric systems, in the series under consideration, it is observed already in the relatively symmetric perfluoromethane (1)-n-butane (2) system,2 where the difference between the critical temperatures of components does not exceed 200 K. In comparison, in the methane (1)-nalkane (2) series, this difference approaches 320 K (the methane (1)-n-hexane (2) system),3 and, in the ethane (1)-n-alkane (2) series, 440 K (the ethane (2)-n-octadecane (2) system).4 Although perfluoromethane (1)-alkane (2) mixtures have been discussed in the literature,5 there is still no convincing explanation for their anomalous phase behavior.6 Thus, the prediction of their phase behavior presents a challenging task for thermodynamic models. Thus far, the most successful attempt to model the phase diagrams of the systems under consideration has been performed by McCabe et al.,6 using the theoretically based statistical associating fluid theory for chain molecules with interaction potentials of variable range (SAFTVR) approach. This model was able to predict the data of the light members of perfluoromethane (1)-n-alkanes (2) series accurately, using the same value of the binary adjustable parameter. However, the results for the heavier homologues have presented deviation from experimental data, which required readjustment of the binary parameter. The present study continues a series of papers7-15 that have examined the ability of the novel Global Phase Diagram-based * Tel.: +972-3-9066346. Fax: +972-3-9066323. E-mail address:
[email protected].
semi-predictive approach (GPDA) to predict experimental data, in comparison with the two most successful approaches, namely, the Predictive Soave-Redlich-Kwong (PSRK)16-20 method and the Linear Combination of the Vidal and Michelsen mixing rules (LCVM).21-25 However, LCVM has not been included in the present study, because its parameter matrix does not incorporate the CF4 group; hence, this time, only the results of GPDA and PSRK are compared. 2. Theory The equation of state (EOS) implemented by GPDA is
P)
RT(Vm + 0.125b) Vm(Vm - 0.875b)
-
2 aTr(m1Tm r )
(Vm + c)(Vm + d)
(1)
The methodology used to evaluate the parameters of eq 1 has been presented in previous studies.7-15 For CF4, m1 ) -0.32114 and m2 ) 0.07488. The mixture parameter values are obtained using the following classical van der Waals mixing rules:
Z)
∑ij XiXjZij
(2)
where z ) a, b, c, and d. The cross-interaction parameters are obtained using the following combination rules:
a21 ) a12 ) (1 - K12)xa11a22
(3a)
b12 ) b21 ) (1 - I12)
(3b)
b11 + b22 2 c11 + c22 c12 ) c21 ) 2 d11 + d22 d12 ) d21 ) 2
10.1021/ie060817m CCC: $33.50 © 2006 American Chemical Society Published on Web 08/30/2006
(3c) (3d)
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where k12 and l12 are binary adjustable parameters whose values are estimated at the intersection of the loci that present the experimental values of CPM and UCEP (or, alternatively, TCP) on the klGPD of the reference homologue:
[
]
Tc2 k12 ) K11 - l12 (1 - t) + K22t Tc1
(4)
where K11 and K22 are characteristic values for a given homologues series and t is given by the following dimensionless functionality:
t ) tanh
[(
T - Tc1 T*c2 - Tc1
)] 2
(5)
For homologues heavier than the reference homologue (CO2n-butane), T* c2 ) Tc2. For lighter homologues, T* c2 is equal to the Tc2 of the reference homologue. Therefore, for all lighter homologues, T*c2 will be taken to be equal to 425.12 K, which is the Tc of n-butane. This will allow the same temperature dependency of k12 to be maintained along the homologues series. For the CF4-alkane series, K11 ) 0.009, K22 ) 0.14, and l12 ) -0.05. In other words, GPDA uses three binary parameters to predict the data of the homologues series; PSRK uses up to six specific series-dependent binary parameters. However, in the present case, only two of them have nonzero values. In the following discussion, we will compare the ability of GPDA and PSRK to predict the phase equilibria data in the systems under consideration. 3. Results Figure 1 compares the critical points of the perfluoromethane (1)-alkane (2) systems that are available in the literature2,26,28 or have been extrapolated from the isopleths27 with the predictions yielded by EOSs, over a wide range of temperatures and pressures. The results presented by PSRK are similar to those described in previous studies,8,9 namely, underestimation of the LLE range for the light members of the series and its overestimation for the heavy members. In particular, it can be observed that PSRK underestimates the temperature of the liquid-liquid split in the perfluoromethane (1)-ethane (2), perfluoromethane (1)-propane (2), and perfluoromethane (1)n-butane (2) systems, although it overestimates the range of vapor-liquid equilibria (VLE) of the last system. In the perfluoromethane (1)-n-heptane (2) system, PSRK substantially overestimates both LLE and VLE, and for the asymmetric systems such as perfluoromethane (1)-n-dodecane (2) and perfluoromethane (1)-squalane (2), the results of PSRK are in total disagreement with the experimental data. At the same time, note that PSRK correctly predicts the transition from Type II to Type III phase behavior between the perfluoromethane (1)propane (2) and perfluoromethane (1)-n-butane (2) systems. In other words, PSRK is able to describe the patterns of phase behavior in the series under consideration qualitatively, but it fails to yield quantitatively accurate results for the critical data. In contrast, GPDA is able to predict the data both qualitatively and quantitatively. The model only exhibits a noticeable deviation from the experimental points for an asymmetric system such as perfluoromethane (1)-squalane (2). In particular, it underestimates the temperatures of the LLE critical points below a pressure of 1000 bar and overestimates them at higher pressures. Such a result can be explained by the fact that the phase behavior in this system is totally different than that in
Figure 1. Critical lines of the perfluoromethane (1)-n-alkane (2) systems. Solid lines represent calculated critical data, dashed lines represent vaporpressure data, and symbols represent experimental data ((1) perfluoromethane (1)-ethane (2),26 (4) perfluoromethane (1)-propane (2),27 (O) perfluoromethane (1)-n-butane (2),2,27 (b) perfluoromethane (1)-n-heptane (2),28 (2) perfluoromethane (1)-n-dodecane (2),28 and (3) perfluoromethane (1)-squalane (2)28).
Figure 2. Liquid-liquid equilibria (LLE) of the perfluoromethane (1)squalane (2) system. Solid lines represent data predicted by GPDA, and symbols represent experimental isopleths28 ((b) x ) 0.797, (O) x ) 0.832, (2) x ) 0.854, and (4) x ) 0.89).
the perfluoromethane (1)-butane (2) system. In the following discussion, the subcritical VLE and LLE data will be considered. Figure 2 presents the LLE data of the perfluoromethane (1)squalane (2) system. The PSRK predictions for this system are totally wrong, and they are located outside the plot. Although the predictions of GPDA are imperfect, it can be observed that this model still gives a reasonable estimation of the data. The same conclusions can be drawn in regard to the results for the perfluoromethane (1)-n-dodecane (2) system (see Figure 3). The isopleths (both VLE and LLE) of the perfluoromethane (1)-n-heptane (2) system are presented in Figure 4. It can be
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Figure 3. LLE of the perfluoromethane (1)-n-dodecane (2) system. Solid lines represent data predicted by GPDA, and symbols represent experimental isopleths28 ((b) x ) 0.757, (O) x ) 0.838, (2) x ) 0.872, and (4) x ) 0.896).
Figure 5. Critical data of the perfluoromethane (1)-n-butane (2) system. Solid lines represent data predicted by GPDA, dotted lines represent data predicted by PSRK, and solid circles (b) represent experimental data.2
Figure 4. LLE of the perfluoromethane (1)-n-dodecane (2) system. Solid lines represent data predicted by GPDA, dotted lines represent data predicted by PSRK, and symbols represent experimental isopleths28 ((b) x ) 0.492, (O) x ) 0.575, (2), x ) 0.637, and (4) x ) 0.809).
observed that the curvature of the isopleths generated by PSRK does not match the experimental data. In particular, PSRK substantially overestimates the VLE pressures. In addition, the temperature of experimental LLE increases with pressure, which PSRK fails to predict. This result can be explained by the fact that PSRK is adjusted to experimental data using only the parameter a, while keeping the value of the binary parameter for covolume (l12) zero. This parameter has a significant influence on the results that are yielded by EOSs. In particular, decreasing l12 increases the range of LLE and directs the liquidliquid isopleths toward high temperatures, which is a behavior that is characteristic for the systems under consideration. Thus, GPDA yields very accurate results; only at x ) 0.492 can a certain deviation from data be detected. Keeping l12 ) 0 directs the critical line toward infinite pressure, which results in its almost-vertical slope, as presented by PSRK, in total disagreement with the experimental data (see Figure 4). Figure 5 demonstrates that GPDA predicts the critical data of the perfluoromethane (1)-n-butane (2) system precisely in both pressure-composition (P-x) and pressure-temperature (P-T) projections. As noticed previously, PSRK is capable of only a qualitative description of data. Again, the results of this model show the characteristic patterns of behavior, namely, the overestimation of VLE and the incorrect slope of LLE, which now results in its substantial underestimation. The same conclusions can be drawn in regard to the subcritical LLE
Figure 6. LLE of the perfluoromethane (1)-n-butane (2) system. Solid lines represent data predicted by GPDA, dotted lines represent data predicted by PSRK, and symbols represent experimental isopleths27 ((b) x ) 0.086, (O) x ) 0.238, (2) x ) 0.339, (4) x ) 0.95).
(Figure 6) and VLE (Figure 7) data of the perfluoromethane (1)-n-butane (2) system, and the LLE data of the perfluoromethane (1)-methylpropane (2) system (Figure 8), the perfluoromethane (1)-propane (2) system (Figure 9), and the perfluoromethane (1)-ethane (2) system (Figure 10). 4. Conclusions The present study continues a series of papers7-15 that have compared the GPDA, PSRK,16-20 and LCVM21-25 EOS models. This time, the mixtures of perfluoromethane (1) and alkanes (2) have been investigated. Because the LCVM parameter matrix does not incorporate the CF4 group, this model has not been considered. The results of the present study are consistent with the previous studies, and they, once again, demonstrate that the multidimensional nature of the GE-based model such as PSRK substantially hinders consideration of the complete thermodynamic phase space, which includes VLE in the entire temper-
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Figure 7. Vapor-liquid equilibria (VLE) of the perfluoromethane (1)n-butane (2) system. Solid lines represent data predicted by GPDA, dotted lines represent data predicted by PSRK, and symbols represent experimental isopleths2 ((3) x ) 0.103, (1) x ) 0.303, (b) x ) 0.624).
Figure 10. LLE of perfluoromethane (1)-ethane (2). Solid lines represent data predicted by GPDA, dotted lines represent data predicted by PSRK, and symbols represent experimental isopleths26 ((b) x ) 0.172, (O) x ) 0.492, (2) x ) 0.59, (4) x ) 0.618).
total disagreement with the experimental data. In addition, because PSRK does not use the binary parameter l12, it generates an incorrect slope of the LLE critical lines and isopleths in P-T projection. In contrast, GPDA predicts robust results, even for very asymmetric systems, and yields a precise description of the more-symmetric systems. These facts support a doubtless advantage of the approach that is based on consideration of the complete phase diagram and not only of its separate parts. Nomenclature
Figure 8. LLE of the perfluoromethane (1)-methylpropane (2) system. Solid lines represent data predicted by GPDA, dotted lines represent data predicted by PSRK, and symbols represent experimental isopleths29 ((b) x ) 0.243, (O) x ) 0.414, (2) x ) 0.8, and (4) x ) 0.921).
a ) cohesion parameter b ) co-volume c, d ) attraction density dependence parameters in eq 1 P ) pressure R ) universal gas constant T ) temperature x ) mole fraction of the lighter compound in the liquid phase V ) volume Subscripts c ) critical state m ) molar property AbbreViations EOS ) equation of state GPDA ) Global Phase Diagram approach klGPD ) Global Phase Diagram in the k12-l12 projection LLE ) liquid-liquid equilibria PSRK ) predictive Soave-Redlich-Kwong group contribution EOS VLE ) vapor-liquid equilibria
Figure 9. LLE of the perfluoromethane (1)-propane (2) system. Solid lines represent data predicted by GPDA, dotted lines represent data predicted by PSRK, and symbols represent experimental isopleths27 ((b) x ) 0.218, (O) x ) 0.344, (2) x ) 0.67, and (4) x ) 0.79).
ature range, LLE, LLV, and critical lines as well. As a result, PSRK does not establish the correct inter-relationship between VLE and LLE. Although, this time, PSRK is able to describe the patterns of phase behavior in the series under consideration, it fails to predict the data quantitatively. In particular, it underestimates the LLE range for the light members of the homologous series and overestimates it for the heavy members. In very asymmetric systems, the predictions of PSRK are in
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ReceiVed for reView June 27, 2006 ReVised manuscript receiVed August 2, 2006 Accepted August 7, 2006 IE060817M
